Preface: As with the other posts about running injuries, go to the doctor if pain persists. This post is not all inclusive and is basic information.
When addressing the core, you have to know what the Transverse Abdominis (TrA) muscle is all about. It is a lumbar stabilizing muscle. Instead of going into detail about it, you can read HERE. In the article, they try to describe how to contract the muscle but it sounds confusing. If the article is confusing to you, just remember: bring your belly button towards your back bone (you want to feel a tightening sensation in your abdominal area) without holding your breath. If you simply suck in air, you are contracting other muscles and not isolating the TrA.
Next, I will give you some exercises that runners (or any worker-outer) can do for core strengthening (you can adjust hold times and number of reps...these are just examples). When you perform these exercises, it's important to first contract the TrA to stabilize the lumbar spine.
-Bridge (hold 5-10 seconds, lower your bum, repeat 10 times)
Another variation (you can decrease reps, maybe 5 on each side):
-Plank (hold 30-60 seconds)
-Side Plank (hold 30-60 seconds)
-Plank with leg lift (5 reps each side holding 5 seconds each)
-Superman/Banana (name from P90X; hold up to 5 seconds each, 5-10 reps)
-Russian Twist (you can hold a weight or go weightless, try 10-20 reps each way...increase reps if it's easy)
-Step Downs (I'm not a fan of the name they gave it...check out the description HERE)
Yes, that's me...and, yes, those are my dogs toys in the background.
-Finally, you can incorporate exercise balls/BOSU//balance discs with other exercises such as with arm workouts (example: stand on BOSU while doing biceps curls).
Again, always contract your TrA before you start and during each exercise!